E.C. 2. (10 pts.) Suppose that (sn) is a sequence of real numbers such that sn...

E.C. 2. (10 pts.) Suppose that (sn) is a sequence of real numbers such that sn ≥ 0 for all n ∈ N. (a) Show that the set of subsequential limits of S satisfies S ⊆ [0,∞) ∪ {+∞}. (b) Is it possible for S = [0,∞) ? (Hint: apply Theorem 11.9.)

Legible handwriting is a must

Expert Solution

Colby Messinger answered 2 years ago

Prove that if a sequence is bounded, then limsup sn is a real number.

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all...

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many sn belong to [a,b], then lim sn belongs to [a, b].

A sequence is just an infinite list of numbers (say real numbers, we often denote these...

A sequence is just an infinite list of numbers (say real numbers, we often denote these by a0,a1,a2,a3,a4,.....,ak,..... so that ak denotes the k-th term in the sequence. It is not hard to see that the set of all sequences, which we will call S, is a vector space. a) Consider the subset, F, of all sequences, S, which satisfy: ∀k ≥ 2,a(sub)k = a(sub)k−1 + a(sub)k−2. Prove that F is a vector subspace of S. b) Prove that if...

Consider the sequence sn defined as: s0 = 1 s1 = 1 sn = 2sn-1 +...

Consider the sequence sn defined as: s0 = 1 s1 = 1 sn = 2sn-1 + sn-2 What is the base case for this recursive relation? Find s5 Write the pseudocode for a recursive function to find Sn for any arbitrary value of n. Create a non-recursive formula for finding the nth term in the sequence in O(1) time.

(a) The Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence,...

(a) The Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and are characterised by the fact that every number after the first two is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 114, … etc. By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. We define Fib(0)=0,...

Let S = (s1, s2, . . . , sn) be a given sequence of integer...

Let S = (s1, s2, . . . , sn) be a given sequence of integer numbers. The numbers can be positive or negative. We define a slice of S as a sub- sequence (si,si+1,...,sj) where 1 ≤ i < j ≤ n. The weight of a slice is defined as the sum of its elements. Provide efficient algorithms to answer each of the following questions: a)Is there any slice with zero weight ? b)Find the maximum weight slice in...

Suppose A is the set of positive real numbers, and suppose u and v are two...

Suppose A is the set of positive real numbers, and suppose u and v are two strictly increasing functions.1 It is intuitive that u and v are ordinally equivalent, since both rank larger numbers higher, and therefore generate the same ranking of numbers. Write this intuition as a proof.

Both TCP and RTP use sequence numbers. Investigate whether or not the sequence numbers in the...

Both TCP and RTP use sequence numbers. Investigate whether or not the sequence numbers in the TCP and RTP protocols play the same role.

Suppose the function g(x) has a domain of all real numbers except x = −2 . The...

Suppose the function g(x) has a domain of all real numbers except x = −2 . The second derivative of g(x) is shown below. g ''(x) = (x−1)(x + 3) (x + 2)3 (a) Give the intervals where g(x) is concave down. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) (b) Give the intervals where g(x) is concave up. (Enter your answer using interval notation. If an answer does not exist, enter DNE.) (c) Find...

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn...

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn where k ≤ n. (a) For σ ∈ Sn, prove that στσ-1 = (σ(1), σ(2), . . . , σ(k)). (b) Let ρ be any cycle of length k in Sn. Prove that there exists an element σ ∈ Sn so that στσ-1 = ρ.

Question